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If elements $a$ and $b$ of a group are of finite order, what can we say about the order of $ab$?

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  • $\begingroup$ Not much, it might be infinite. Obviously, you have to look at non-commutative groups to find an example. $\endgroup$ Jun 4, 2016 at 9:24

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If $a$ and $b$ commutes, i.e. $ab=ba$

If $a^n=b^m=e$, then $(ab)^{\gcd{m,n}}=a^{\gcd(m,n)}b^{\gcd(m,n)}=e$.

Counterexample for the noncommutative case

Let $G$ be the group of rank $2$ with the additional relations $a^2=b^2=1$. Then, $(ab)^n=abababab\ldots ab$ is a different non-zero element for every $n$, thus, the order of $ab$ is infinite.

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